Limited-Memory BFGS Diagonal Preconditioners for a Data Assimilation Problem in Meteorology
Limited-Memory BFGS Diagonal Preconditioners for a Data Assimilation Problem in Meteorology
复制标题
用于气象数据同化问题的有限内存 BFGS 对角预处理器
DOI：
10.1023/a:1010030224033
复制
发表时间：
2000
影响因子：
2.1
通讯作者：
Matthew Fisher
中科院分区：
文献类型：
--
作者：
F. Veersé;D. Auroux;Matthew Fisher
本文利用周期域上一维粘性Burgers型方程的两个简单的变分同化问题，研究了不同的对角预条件更新和尺度策略对有限记忆BFGS(Broyden，Fletcher，Goldfarb和Shanno)逆Hessian近似和最小化性能的影响。这些简单的问题与气象学和海洋学中常见的大尺度变分同化问题有一些共同的特点，所研究的更新公式是Gilbert和Lemaréchal(Math)提出的公式。Prog.，Vol.45，pp.407-435,1989)和朱等人的准柯西公式。(暹罗期刊，第9卷，第1192-1204页，1999)。首先考虑应该使用哪种信息来更新对角线预处理器，是要忘记的信息还是最近的信息。然后，在前人的基础上，针对相应的公式引入了对角预条件子的比例变换，以改善极小化性能。这种标度对L-BFGS逆海森近似的质量有很大的负面影响，这使得我们提出了一种交替的更新和标度策略，它提供了一个很好的逆海森近似，并对所考虑的问题给出了最好的最小化性能。通过这种方法，逆Hessian近似的质量在最小化过程中稳步提高。此外，这种质量和L-BFGS最小化性能随着存储信息量的增加而提高。
This paper uses two simple variational data assimilation problems with the 1D viscous Burgers' equation on a periodic domain to investigate the impact of various diagonal-preconditioner update and scaling strategies, both on the limited-memory BFGS (Broyden, Fletcher, Goldfarb and Shanno) inverse Hessian approximation and on the minimization performance. These simple problems share some characteristics with the large-scale variational data assimilation problems commonly dealt with in meteorology and oceanography.The update formulae studied are those proposed by Gilbert and Lemaréchal (Math. Prog., vol. 45, pp. 407–435, 1989) and the quasi-Cauchy formula of Zhu et al. (SIAM J. Optim., vol. 9, pp. 1192–1204, 1999). Which information should be used for updating the diagonal preconditioner, the one to be forgotten or the most recent one, is considered first. Then, following the former authors, a scaling of the diagonal preconditioner is introduced for the corresponding formulae in order to improve the minimization performance. The large negative impact of such a scaling on the quality of the L-BFGS inverse Hessian approximation led us to propose an alternate updating and scaling strategy, that provides a good inverse Hessian approximation and gives the best minimization performance for the problems considered. With this approach the quality of the inverse Hessian approximation improves steadily during the minimization process. Moreover, this quality and the L-BFGS minimization performance improves when the amount of stored information is increased.